#6616958

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(17,18,23,27)(19,28,31,39)(20,32,33,25)(21,26,22,36)(24,37,40,30)(29,43,44,42)(34,47,35,41)(38,45,46,48);(1,3,5,4)(2,6,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,29)(21,30)(22,37)(23,31)(25,38)(26,39)(27,40)(28,36)(32,46)(33,44)(34,45)(35,48)(41,43)(42,47);(1,3,5,4)(2,7,8,6)(10,13)(11,15)(17,20,23,33)(18,25,27,32)(19,29,31,44)(21,34,22,35)(24,38,40,46)(26,41,36,47)(28,42,39,43)(30,45,37,48);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(20,33)(25,32)(29,44)(34,35)(38,46)(41,47)(42,43)(45,48);(1,4,5,3)(2,7,8,6)(10,13)(11,15)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,34,33,35)(24,39,40,28)(25,41,32,47)(29,45,44,48)(38,43,46,42)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=gi,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6524154);